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                <div class="container"><article class="page"><h1 class="post-title animated flipInX">代数学习笔记</h1><div class="post-meta">
            <div class="post-meta-main"><a class="author" href="/" rel="author" target="_blank">
                    <i class="fas fa-user-circle fa-fw"></i>ChenDong Zhu
                </a>&nbsp;<span class="post-category">收录于&nbsp;<i class="far fa-folder fa-fw"></i><a href="/categories/%E5%AD%A6%E4%B9%A0/">学习</a>&nbsp;</span></div>
            <div class="post-meta-other"><i class="far fa-calendar-alt fa-fw"></i><time datetime=2021-09-22>2021-09-22</time>&nbsp;
                <i class="fas fa-pencil-alt fa-fw"></i>约 271 字&nbsp;
                <i class="far fa-clock fa-fw"></i>预计阅读 1 分钟&nbsp;</div>
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                <h2 class="post-toc-title">目录</h2>
                <div class="post-toc-content"><nav id="TableOfContents">
  <ul>
    <li><a href="#背景">背景</a></li>
  </ul>

  <ul>
    <li><a href="#ch-13-functions">Ch 1.3 Functions</a></li>
  </ul>
</nav></div>
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                    <div class="post-toc-content"><nav id="TableOfContentsMobile">
  <ul>
    <li><a href="#背景">背景</a></li>
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  <ul>
    <li><a href="#ch-13-functions">Ch 1.3 Functions</a></li>
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            </div><div class="post-content"><a class="post-dummy-target" id="背景"></a><h2>背景</h2>
<p>使用教材：<em><strong>Algebra by Thomas W. Hungerford</strong></em></p>
<h1 id="ch-1-introductin">Ch 1 Introductin</h1>
<p><strong>relative complement</strong></p>
<p>The relative complement of A in B is the following subclass of B:
$$
B - A = {x|x\in B\ and\ x\notin A}
$$</p>
<a class="post-dummy-target" id="ch-13-functions"></a><h2>Ch 1.3 Functions</h2>
<p>Given classes $A$ and $B$, a <strong>function</strong> (or <strong>map</strong> or <strong>mapping</strong>) $f$ from $A$ to $B$ :
$$
f:A \rightarrow B
$$
$A$ is the <strong>domain</strong> and $B$ is the <strong>codomain</strong> or <strong>range</strong></p>
<p>Sometime it is convenient to denote the effect of the function $f$ on an element of $A$ by
$$
a \mapsto f(a)
$$
if $S \subset A$, the function from $S$ to $B$ given by
$$
a \mapsto f(a), for\ a \in S
$$
is called the <strong>restriction</strong> of $f$ to $S$ and is denoted $f|S:S\rightarrow B$.</p>
<p>If $A$ is any class, the <strong>identity function</strong> on $A$(denoted $1_A:A\rightarrow A$) is the function given by $a\mapsto a$. If $S \subset A$,</p>
<p>the function $1_A|S:S\rightarrow A$ is called the <strong>inclusion map</strong> of $S$ into A.</p>
<p>The <strong>composite</strong> of $f$ and $g$ is the function $A \rightarrow C$ given by
$$
a \mapsto g(f(a)),\ a\in A
$$
The compostie function is denoted $g\ \circ \ f$ or simply $gf$.</p>
<p>A <strong>diagram</strong> of functions:</p>
<p><figure><img src="/svg/loading.min.svg" data-sizes="auto" data-src="/images/algebra/0001.png" alt="" class="lazyload"></figure></p>
<p>is said to be <strong>commutative</strong> if $gf = h$.</p>
<p>If a diagram is said to be commutative if every triangle and square in it is commutative.</p>
<p>Let $f:A \rightarrow B$ be a function. If $S \sub A$, the <strong>image of S under f</strong> (denoted $f(S)$) is the class
$$
{b\in B\ |\ b=f(a)\ for\ some\ a\in S}
$$</p>
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